docs(arap): B-41, the bookspec examples, and the four registers (P4b S7)

rules.md B-41 states the write-off: the direct method the owner chose
(A-5), Dr 6990 / Cr 1200 with the receivable tagged, at most what is
owed and at most the least owed from that day on (a partial write-off as
§1.166-3(a) allows), refused on cash-basis books (§1.166-1(e)), its
evidence filed in the same call (§1.166-2(a)); bills are never written
off (ASC 405-20-40-1); and a recovery is not a new movement (D-49): the
same year, reverse the write-off and record the payment; a later year, a
plain entry to a bad-debt-recoveries revenue account, the invoice staying
written off (IRC §111).

bookspec states B-41 for the first time and gives B-38 its example: two
new examples, bad-debt-direct-write-off (two partial write-offs of one
invoice, the balances and the trial balance worked by hand) and
arap-hand-control-posting-refused (a hand line an earlier version wrote
replays, the same line written today is refused). The example schema
gains the dated write_off event, lets a refused entry say so (an entry
posts whole or not at all; older and refused never combine), and its
refusal vocabulary gains untagged_control_line, B-38's control guard.

The registers: docs/architecture.md names the new tool group;
docs/receipt-store.md lists invoice.written_off and what replay does with
it; docs/parity.md records the slice (no row changes status, the floor
stands); docs/testing.md records every test and what P12 must mutate;
docs/verification.md the two Lean theorems and what they cover.

No figure moves: no golden file changed.

spec/rules-index.md

@@ -6,8 +6,8 @@ what its statement rests on and the examples that demonstrate it.
 
 A rule whose only source is a human choice carries rules.md's **CHOICE**, bookspec's status of the same name:
 the specification names the options, an engine declares its pick
-(`claims.choices`, `bookspec_checker::Claims`). Rules B-39 to B-45 are
-allocated in `plan/served.md` 8.0.3 and not yet stated: their
+(`claims.choices`, `bookspec_checker::Claims`). Rules B-39, B-40 and B-42
+to B-45 are allocated in `plan/served.md` 8.0.3 and not yet stated: their
 identifier and area are fixed here so a later unit fills them in
 without moving anyone else's number.
 
@@ -50,10 +50,10 @@ without moving anyone else's number.
 | B-35 | `BS-partner-3` | partner | CONVENTION | A party's part of an entry, and a party's totals | `p3-party-accrual-counts-once`, `p3-party-distribution-still-owed`, `p3-party-gift-card-is-a-promise`, `p3-party-members-equity-and-k1`, `p3-party-own-liability-never-pays`, `p3-party-reversed-card-payment`, `p3-party-signs-and-reversals` |
 | B-36 | `BS-partner-4` | partner | AUTHORITY | The K-1 worksheet reports tax-basis capital beside book capital | `partnership-k1-tax-basis-capital`, `partnership-k1-tax-basis-odd-cents` |
 | B-37 | `BS-statement-5` | statement | CHOICE | A project is a class with its facts; its figures are its tagged lines | `projects-invoice-and-bill-lines-count-by-project`, `projects-net-is-its-tagged-lines`, `projects-on-spec-costs-are-expenses` |
-| B-38 | `BS-arap-2` | arap | CHOICE | A document's figures are its ledger lines that stand; a control account is its documents | — |
+| B-38 | `BS-arap-2` | arap | CHOICE | A document's figures are its ledger lines that stand; a control account is its documents | `arap-hand-control-posting-refused` |
 | B-39 | `BS-arap-3` | arap | — | Terms: when a document is due (allocated, not yet stated) | — |
 | B-40 | `BS-arap-4` | arap | — | Numbers and the issued copy (allocated, not yet stated) | — |
-| B-41 | `BS-arap-5` | arap | — | Write-offs (allocated, not yet stated) | — |
+| B-41 | `BS-arap-5` | arap | CHOICE | A debt that will not be collected is written off, and a recovery is income | `bad-debt-direct-write-off` |
 | B-42 | `BS-arap-6` | arap | — | Settlements and the fee (allocated, not yet stated) | — |
 | B-43 | `BS-entry-5` | entry | — | Drafts and evidence (allocated, not yet stated) | — |
 | B-44 | `BS-ledger-12` | ledger | — | Extraction and dedup (allocated, not yet stated) | — |